| $Coin$ | $+15$ | $+12$ | $-6$ |
| $Probability$ | $\frac{6}{{36}}$ | $\frac{4}{{36}}$ | $\frac{{26}}{{36}}$ |
Probability of doublet $=\frac{6}{36}$
Probability of sum of $9=\frac{4}{36}$
Other probability $=\frac{26}{36}$
Expected gain/loss $=15 \times \frac{6}{36}+12 \times \frac{4}{36}-6 \times \frac{26}{36}$
$=\frac{90}{36}+\frac{48}{36}-\frac{156}{36}=\frac{-1}{2}$
Hence correct answer is option $(D)$
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$\sin \left(2 x^{2}\right) \log _{c}\left(\tan x^{2}\right) d y+\left(4 x y-4 \sqrt{2} x \sin \left(x^{2}-\frac{\pi}{4}\right)\right) d x=0$
$0 < x < \sqrt{\frac{\pi}{2}}$, which passes through the point $\left(\sqrt{\frac{\pi}{6}}, 1\right)$. Then $\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|$ is equal to $.....$